Shock waves driven in the intergalactic medium during the merging of galaxy clusters have been observed in X-ray imaging and spectroscopy. Fluid motions inferred from the shock strength and morphology can be compared to the cold dark matter (CDM) distribution inferred from gravitational lensing. A detailed reconstruction of the CDM kinematics, however, must take into account the nontrivial response of the fluid intracluster medium to the collisionless CDM motions. We have carried out two-dimensional simulations of gas dynamics in cluster collisions. We analyze the relative motion of the clusters, the bow shock wave, and the contact discontinuity, and relate these to X-ray data. We focus on the Bullet Cluster, 1E 0657-56, a near-head-on collision of unequal-mass clusters, for which the gas density and temperature jumps across the prominent bow shock imply a high shock velocity, 4700 km s -1 . The velocity of the fluid shock has been widely interpreted as the relative velocity of the CDM components. This need not be the case, however. An illustrative simulation finds that the present relative velocity of the CDM halos is ~16% lower than that of the shock. While this conclusion is sensitive to the detailed initial mass and gas density profile of the colliding clusters, such a decrease of the inferred halo relative velocity would increase the likelihood of finding 1E 0657-56 in a ΛCDM universe.